research

Book

This is a preliminary version of a book which presents the quantitative homogenization and large-scale regularity theory for elliptic equations in divergence-form. The self-contained presentation gives new and simplified proofs of the core results proved in the last several years, including the algebraic convergence rate for the variational subadditive quantities, the large-scale Lipschitz and higher regularity estimates and Liouville-type results, optimal quantitative estimates on the first-order correctors and their scaling limit to a Gaussian free field. The last chapter contains new results on the homogenization of the Dirichlet problem, including optimal quantitative estimates of the homogenization error and the two-scale expansion.

Papers

We argue that Hamilton-Jacobi equations provide a convenient and intuitive approach for studying the large-scale behavior of mean-field disordered systems. This point of view is illustrated on the problem of inference of a rank-one matrix. We compute the large-scale limit of the free energy by showing that it satisfies an approximate Hamilton-Jacobi equation with asymptotically vanishing viscosity parameter and error term.

We introduce a new iterative method for computing solutions of elliptic equations with random rapidly oscillating coefficients. Similarly to a multigrid method, each step of the iteration involves different computations meant to address different length scales. However, we use here the homogenized equation on all scales larger than a fixed multiple of the scale of oscillation of the coefficients. While the performance of standard multigrid methods degrades rapidly under the regime of large scale separation that we consider here, we show an explicit estimate on the contraction factor of our method which is independent of the size of the domain. We also present numerical experiments which confirm the effectiveness of the method.

We develop a quantitative theory of stochastic homogenization for linear, uniformly parabolic equations with coefficients depending on space and time. Inspired by recent works in the elliptic setting, our analysis is focused on certain subadditive quantities derived from a variational interpretation of parabolic equations. These subadditive quantities are intimately connected to spatial averages of the fluxes and gradients of solutions. We implement a renormalization-type scheme to obtain an algebraic rate for their convergence, which is essentially a quantification of the weak convergence of the gradients and fluxes of solutions to their homogenized limits. As a consequence, we obtain estimates of the homogenization error for the Cauchy-Dirichlet problem which are optimal in stochastic integrability. We also develop a higher regularity theory for solutions of the heterogeneous equation, including a uniform C0,1-type estimate and a Liouville theorem of every finite order.

We aim to give a pedagogic and essentially self-contained presentation of the construction of various stochastic objects appearing in the dynamical 𝜑43model. The construction presented here is based on the use of paraproducts. The emphasis is on describing the stochastic objects themselves rather than introducing a solution theory for the equation.

We introduce new methods to compute the homogenized coefficients of divergence-form operators with random coefficients. We focus on a discrete-space setting with i.i.d. coefficients, and investigate algorithms which take a sample of the random coefficient field as input. In order to produce an approximation of the homogenized coefficients at precision δ, any algorithm must perform at least of the order of δ-2 operations. We present an algorithm that essentially achieves this lower bound, up to logarithmic factors. This improves upon the previously best known method by a factor of δ-1/d. An additional new feature is that the method is cumulative: all computations done at a coarse precision remain useful if the estimate needs to be refined.

We show a diffusive upper bound on the transition probability of a tagged particle in the symmetric simple exclusion process. The proof relies on optimal spectral gap estimates for the dynamics in finite volume, which are of independent interest. We also show off-diagonal estimates of Carne-Varopoulos type.

We prove quantitative estimates on the rate of convergence for the oscillating Dirichlet problem in periodic homogenization of divergence-form uniformly elliptic systems. The estimates are optimal in dimensions larger than three (at least) and new in every dimension. We also prove a regularity estimate on the homogenized boundary condition.

Many real-world networks of interest are embedded in physical space. We present a new random graph model aiming to reflect the interplay between the geometries of the graph and of the underlying space. The model favors configurations with small average graph distance between vertices, but adding an edge comes at a cost measured according to the geometry of the ambient physical space. In most cases, we identify the order of magnitude of the average graph distance as a function of the parameters of the model. As the proofs reveal, hierarchical structures naturally emerge from our simple modeling assumptions. Moreover, a critical regime exhibits an infinite number of discontinuous phase transitions.

We study discrete linear divergence-form operators with random coefficients, also known as the random conductance model. We assume that the conductances are bounded, independent and stationary; the law of a conductance may depend on the orientation of the associated edge. We give a simple necessary and sufficient condition for the relaxation of the environment seen by the particle to be diffusive, in the sense of every polynomial moment. As a consequence, we derive polynomial moment estimates on the corrector.

One of the principal difficulties in stochastic homogenization is transferring quantitative ergodic information from the coefficients to the solutions, since the latter are nonlocal functions of the former. In this paper, we address this problem in a new way, in the context of linear elliptic equations in divergence form, by showing that certain quantities associated to the energy density of solutions are essentially additive. As a result, we are able to prove quantitative estimates on the weak convergence of the gradients, fluxes and energy densities of the first-order correctors (under blow-down) which are optimal in both scaling and stochastic integrability. The proof of the additivity is a bootstrap argument, completing the program initiated in our previous work: using the regularity theory recently developed for stochastic homogenization, we reduce the error in additivity as we pass to larger and larger length scales. In the second part of the paper, we use the additivity to derive central limit theorems for these quantities by a reduction to sums of independent random variables. In particular, we prove that the first-order correctors converge, in the large-scale limit, to a variant of the Gaussian free field.

We study a generalization of the notion of Gaussian free field (GFF). Although the extension seems minor, we first show that a generalized GFF does not satisfy the spatial Markov property, unless it is a classical GFF. In stochastic homogenization, the scaling limit of the corrector is a possibly generalized GFF described in terms of an "effective fluctuation tensor" that we denote by Q. We prove an expansion of Q in the regime of small ellipticity ratio. This expansion shows that the scaling limit of the corrector is not necessarily a classical GFF, and in particular does not necessarily satisfy the Markov property.

We prove an a priori bound for the dynamic 𝜑43model on the torus which is independent of the initial condition. In particular, this bound rules out the possibility of finite time blow-up of the solution. It also gives a uniform control over solutions at large times, and thus allows to construct invariant measures via the Krylov-Bogoliubov method. It thereby provides a new dynamic construction of the Euclidean 𝜑43field theory on finite volume. Our method is based on the local-in-time solution theory developed recently by Gubinelli, Imkeller, Perkowski and Catellier, Chouk. The argument relies entirely on deterministic PDE arguments (such as embeddings of Besov spaces and interpolation), which are combined to derive energy inequalities.

We introduce a new method for obtaining quantitative results in stochastic homogenization for linear elliptic equations in divergence form. Unlike previous works on the topic, our method does not use concentration inequalities (such as Poincaré or logarithmic Sobolev inequalities in the probability space) and relies instead on a higher (Ck, k ≥ 1) regularity theory for solutions of the heterogeneous equation, which is valid on length scales larger than a certain specified mesoscopic scale. This regularity theory, which is of independent interest, allows us to, in effect, localize the dependence of the solutions on the coefficients and thereby accelerate the rate of convergence of the expected energy of the cell problem by a bootstrap argument. The fluctuations of the energy are then tightly controlled using subadditivity. The convergence of the energy gives control of the scaling of the spatial averages of gradients and fluxes (that is, it quantifies the weak convergence of these quantities) which yields, by a new "multiscale" Poincaré inequality, quantitative estimates on the sublinearity of the corrector.

We introduce anchored versions of the Nash inequality. They allow to control the L2 norm of a function by Dirichlet forms that are not uniformly elliptic. We then use them to provide heat kernel upper bounds for diffusions in degenerate static and dynamic random environments. As an example, we apply our results to the case of a random walk with degenerate jump rates that depend on an underlying exclusion process at equilibrium.

We investigate the global fluctuations of solutions to elliptic equations with random coefficients in the discrete setting. In dimension d ≥ 3 and for i.i.d. coefficients, we show that after a suitable scaling, these fluctuations converge to a Gaussian field that locally resembles a (generalized) Gaussian free field. The paper begins with a heuristic derivation of the result, which can be read independently and was obtained jointly with Scott Armstrong.

In the homogenization of divergence-form equations with random coefficients, a central role is played by the corrector. We focus on a discrete space setting and on dimension 3 and more. Completing the argument started in previous work, we identify the scaling limit of the corrector, which is akin to a Gaussian free field.

We present a criterion for a family of random distributions to be tight in local Hölder and Besov spaces of possibly negative regularity. We then apply this criterion to the magnetization field of the two-dimensional Ising model at criticality, answering a question of Camia, Garban and Newman.

We show global well-posedness of the dynamic 𝜑4 model in the plane. The model is a non-linear stochastic PDE that can only be interpreted in a "renormalised" sense. Solutions take values in suitable weighted Besov spaces of negative regularity.

We prove an optimal diffusive decay of the environment viewed by the
particle in random walk among random independent conductances, with,
as a main assumption, finite second moment of the conductance. Our
proof, using the analytic approach of Gloria, Neukamm and Otto, is
very short and elementary.

We develop a higher regularity theory for general quasilinear elliptic equations and systems in divergence form with random coefficients. The main result is a large-scale L∞-type estimate for the gradient of a solution. The estimate is proved with optimal stochastic integrability under a one-parameter family of mixing assumptions, allowing for very weak mixing with non-integrable correlations to very strong mixing (e.g., finite range of dependence). We also prove a quenched L2 estimate for the error in homogenization of Dirichlet problems. The approach is based on subadditive arguments which rely on a variational formulation of general quasilinear divergence-form equations.

We investigate the first-order correction in the homogenization of linear parabolic equations with random coefficients. In dimension 3 and higher and for coefficients having a finite range of dependence, we prove a pointwise version of the two-scale expansion. A similar expansion is derived for elliptic equations in divergence form.
The result is surprising, since it was not expected to be true without further symmetry assumptions on the law of the coefficients.

The Ising-Kac model is a variant of the ferromagnetic Ising model in which each spin variable interacts with all spins in a neighbourhood of radius 𝛾 -1 for 𝛾 ≪ 1 around its base point. We study the Glauber dynamics for this model on a discrete two-dimensional torus ℤ2/(2N+1)ℤ2, for a system size N ≫ 𝛾 -1 and for an inverse temperature close to the critical value of the mean field model. We show that the suitably rescaled coarse-grained spin field converges in distribution to the solution of a non-linear stochastic partial differential equation.
This equation is the dynamic version of the 𝜑42quantum field theory, which is formally given by a reaction diffusion equation driven by an additive space-time white noise. It is well-known that in two spatial dimensions, such equations are distribution valued and a Wick renormalisation has to be performed in order to define the non-linear term. Formally, this renormalisation corresponds to adding an infinite mass term to the equation. We show that this need for renormalisation for the limiting equation is reflected in the discrete system by a shift of the critical temperature away from its mean field value.

We consider the contact process with infection rate λ on a random (d+1)-regular graph with n vertices, Gn. We study the extinction time τ (that is, the random amount of time until the infection disappears) as n is taken to infinity. We establish a phase transition depending on whether λ is smaller or larger than λ1(Td), the lower critical value for the contact process on the infinite, (d+1)-regular tree: if λ < λ1(Td), τ grows logarithmically with n, while if λ > λ1(Td), it grows exponentially with n. This result differs from the situation where, instead of Gn, the contact process is considered on the d-ary tree of finite height, since in this case, the transition is known to happen instead at the upper critical value for the contact process on Td.

We consider the simple random walk on ℤd evolving in a random i.i.d. potential taking values in [0,+ ∞). The potential is not assumed integrable, and can be rescaled by a multiplicative factor λ > 0. Completing the work started in a companion paper, we give the asymptotic behaviour of the Lyapunov exponents for d ≥ 3, both annealed and quenched, as the scale parameter λ tends to zero.

We consider the contact process with infection rate λ on the d-ary tree of height n. We study the extinction time τ, that is, the random time it takes for the infection to disappear when the process is started from full occupancy. We prove two conjectures of Stacey regarding τ. Let λ2 denote the upper critical value for the contact process on the infinite d-ary tree. First, if λ < λ2, then τ divided by the height of the tree converges in probability, as n→∞, to a positive constant. Second, if λ > λ2, then log E[τ] divided by the volume of the tree converges in probability to a positive constant, and τ/E[τ] converges in distribution to the exponential distribution of mean 1.

Recently, the quantification of errors in the stochastic homogenization of divergence-form operators has witnessed important progress. Our aim now is to go beyond error bounds, and give precise descriptions of the effect of the randomness, in the large-scale limit. This paper is a first step in this direction. Our main result is to identify the correlation structure of the corrector, in dimension 3 and higher. This correlation structure is similar to, but different from that of a Gaussian free field.

Significance level and positivity bias as causes for high rate of non-reproducible scientific results? [abs] [pdf]

The high fraction of published results that turn out to be incorrect is a major concern of today's science. This paper contributes to the understanding of this problem in two independent directions. First, Johnson's recent claim that hypothesis testing with a significance level of α = 0.05 can alone lead to an unacceptably large proportion of false positive results is shown to be unfounded. Second, a way to quantify the effect of "positivity bias" (the tendency to consider only positive results as worthwhile) is introduced. We estimate the proportion of false positive results among positive results in terms of the significance level used and the positivity ratio. The latter quantity is the fraction of positive results over all results, be they positive or not, published or not. In particular, if one uses a significance level of α = 0.05, and produces 4 (possibly unpublished) negative results for every positive result, then the proportion of false positives among positive results can climb to a high 21%.

Divergence-form operators with stationary random coefficients homogenize over large scales. We investigate the effect of certain perturbations of the medium on the homogenized coefficients. The perturbations that we consider are rare at the local level, but when occurring, have an effect of the same order of magnitude as the initial medium itself. The main result of the paper is a first-order expansion of the homogenized coefficients, as a function of the perturbation parameter.

We show aging of Glauber-type dynamics on the random energy model, in the sense that we obtain the scaling limits of the clock process and of the age process. The latter encodes the Gibbs weight of the configuration occupied by the dynamics. Both limits are expressed in terms of stable subordinators.

This article is concerned with numerical methods to approximate effective coefficients in stochastic homogenization of discrete linear elliptic equations, and their numerical analysis — which has been made possible by recent contributions on quantitative stochastic homogenization theory by two of us and by Otto.
This article makes the connection between our theoretical results and computations.
We give a complete picture of the numerical methods found in the literature, compare them in terms of known (or expected) convergence rates, and study them numerically.
Two types of methods are presented: methods based on the corrector equation, and methods based on random walks in random environments.
The numerical study confirms the sharpness of the analysis (which it completes by making precise the prefactors, next to the convergence rates), supports some of our conjectures, and calls for new theoretical developments.

We consider the simple random walk on ℤd, d ≥ 3, evolving in a potential of the form βV, where (V(x), x ∈ ℤd) are i.i.d. random variables taking values in [0, + ∞), and β > 0. When the potential is integrable, the asymptotic behaviours as β tends to 0 of the associated quenched and annealed Lyapunov exponents are known (and coincide). Here, we do not assume such integrability, and prove a sharp lower bound on the annealed Lyapunov exponent for small β. The result can be rephrased in terms of the decay of the averaged Green function of the Anderson Hamiltonian -Δ + βV.

The article begins with a quantitative version of the martingale central limit theorem, in terms of the Kantorovich distance. This result is then used in the study of the homogenization of discrete parabolic equations with random i.i.d. coefficients. For smooth initial condition, the rescaled solution of such an equation, once averaged over the randomness, is shown to converge polynomially fast to the solution of the homogenized equation, with an explicit exponent depending only on the dimension. Polynomial rate of homogenization for the averaged heat kernel, with an explicit exponent, is then derived. Similar results for elliptic equations are also presented.

We study the extinction time τ of the contact process on finite trees of bounded degree. We show that, if the infection rate is larger than the critical rate for the contact process on ℤ, then, uniformly over all trees of degree bounded by a given number, the expectation of τ grows exponentially with the number of vertices. Additionally, for any sequence of growing trees of bounded degree, τ divided by its expectation converges in distribution to the unitary exponential distribution. These also hold if one considers a sequence of graphs having spanning trees with uniformly bounded degree. Using these results, we consider the contact process on a random graph with vertex degrees following a power law.
Improving a result of Chatterjee and Durrett (2009), we show that, for any infection rate, the extinction time for the contact process on this graph grows exponentially with the number of vertices.

We consider the simple random walk on ℤd evolving in a potential of independent and identically distributed random variables taking values in [0, + ∞]. We give optimal conditions for the existence of the quenched point-to-point Lyapunov exponent, and for different versions of a shape theorem. The method of proof applies as well to first-passage percolation, and builds up on an approach of Cox and Durrett (1981). The weakest form of shape theorem holds whenever the set of sites with finite potential percolates. Under this condition, we then show the existence of the quenched point-to-hyperplane Lyapunov exponent, and give a large deviation principle for the walk under the quenched weighted measure.

We consider the random walk among random conductances on ℤd. We assume that the conductances are independent, identically distributed and uniformly bounded away from 0 and infinity. We obtain a quantitative version of the central limit theorem for this random walk, which takes the form of a Berry-Esseen estimate with speed t-1/10 for d ≤ 2, and speed t-1/5 for d ≥ 3, up to logarithmic corrections.

Consider a discrete-time martingale, and let V2 be its normalized quadratic variation. As V2 approaches 1 and provided some Lindeberg condition is satisfied, the distribution of the rescaled martingale approaches the Gaussian distribution. For any p ≥ 1, Haeusler (1988) gives a bound on the rate of convergence in this central limit theorem, that is the sum of two terms, say Ap + Bp, where up to a constant, Ap = ‖V2-1‖pp/(2p+1). We discuss here the optimality of this term, focusing on the restricted class of martingales with bounded increments. In this context, Bolthausen (1982) sketches a strategy to prove optimality for p = 1. Here, we extend this strategy to any p ≥ 1, thus justifying the optimality of the term Ap. As a necessary step, we also provide a new bound on the rate of convergence in the central limit theorem for martingales with bounded increments that improves on the term Bp, generalizing another result of Bolthausen (1982).

This article is devoted to the analysis of a Monte-Carlo method to approximate effective coefficients in stochastic homogenization of discrete elliptic equations. We consider the case of independent and identically distributed coefficients, and adopt the point of view of the random walk in a random environment. Given some final time t > 0, a natural approximation of the homogenized coefficients is given by the empirical average of the final squared positions rescaled by t of n independent random walks in n independent environments. Relying on a quantitative version of the Kipnis-Varadhan theorem combined with estimates of spectral exponents obtained by an original combination of pde arguments and spectral theory, we first give a sharp estimate of the error between the homogenized coefficients and the expectation of the rescaled final position of the random walk in terms of t. We then complete the error analysis by quantifying the fluctuations of the empirical average in terms of n and t, and prove a large-deviation estimate. Our estimates are optimal, up to a logarithmic correction in dimension 2.

We consider the model of a directed polymer pinned to a line of i.i.d. random charges, and focus on the interior of the delocalized phase. We first show that in this region, the partition function remains bounded. We then prove that for almost every environment of charges, the probability that the number of contact points in [0,n] exceeds c log(n) tends to 0 as n tends to infinity. Our proofs rely on recent results of Birkner, Greven, den Hollander (2010) and Cheliotis, den Hollander (2010).

This article deals with the numerical approximation of effective coefficients in stochastic homogenization of discrete linear elliptic equations. The originality of this work is the use of a well-known abstract spectral representation formula to design and analyze effective and computable approximations of the homogenized coefficients. In particular, we show that information on the edge of the spectrum of the generator of the environment viewed by the particle projected on the local drift yields bounds on the approximation error, and conversely. Combined with results by Otto and the first author in low dimension, and results by the second author in high dimension, this allows us to prove that for any dimension, there exists an explicit numerical strategy to approximate homogenized coefficients which converges at the rate of the central limit theorem.

Attributing a positive value τx to each x in ℤd, we investigate a nearest-neighbour random walk which is reversible for the measure with weights (τx), often known as "Bouchaud's trap model". We assume that these weights are independent, identically distributed and non-integrable random variables (with polynomial tail), and that d ≥ 5. We obtain the quenched subdiffusive scaling limit of the model, the limit being the fractional kinetics process. We begin our proof by expressing the random walk as a time change of a random walk among random conductances. We then focus on proving that the time change converges, under the annealed measure, to a stable subordinator. This is achieved using previous results concerning the mixing properties of the environment viewed by the time-changed random walk.

For the random walk among random conductances, we prove that the environment viewed by the particle converges to equilibrium polynomially fast in the variance sense, our main hypothesis being that the conductances are bounded away from zero. The basis of our method is the establishment of a Nash inequality, followed either by a comparison with the simple random walk or by a more direct analysis based on a martingale decomposition. As an example of application, we show that under certain conditions, our results imply an estimate of the speed of convergence of the mean square displacement of the walk towards its limit.

Let (τx, x ∊ ℤd) be i.i.d. random variables with heavy (polynomial) tails. Given a ∊ [0,1], we consider the Markov process defined by the jump rates τx-(1-a) τya between two neighbours x and y in ℤd. We give the asymptotic behaviour of the principal eigenvalue of the generator of this process, with Dirichlet boundary condition. The prominent feature is a phase transition that occurs at some threshold depending on the dimension.